Place Value, Ordering and Rounding

This lesson covers the fundamental number skills required for AQA GCSE Mathematics. Understanding place value, ordering numbers correctly and rounding to appropriate degrees of accuracy underpins almost every other topic in the specification. These skills appear frequently in both non-calculator and calculator papers.

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Place Value in Integers

Every digit in a whole number has a place value depending on its position. The place value system is based on powers of 10.

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Units
1,000,000	100,000	10,000	1,000	100	10	1

For example, in the number 4,362,517:

• The 4 is worth 4,000,000 (four million)
• The 3 is worth 300,000 (three hundred thousand)
• The 6 is worth 60,000 (sixty thousand)
• The 2 is worth 2,000 (two thousand)
• The 5 is worth 500 (five hundred)
• The 1 is worth 10 (ten)
• The 7 is worth 7 (seven)

Exam Tip: When a question asks "what is the value of the digit 5 in the number 358,201?", they want the full value (50,000), not just the column name (ten thousands).

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Place Value in Decimals

The place value system extends to the right of the decimal point using negative powers of 10.

Units	.	Tenths	Hundredths	Thousandths
1	.	0.1	0.01	0.001

In the number 7.346:

• The 7 is worth 7 units
• The 3 is worth 3 tenths (0.3)
• The 4 is worth 4 hundredths (0.04)
• The 6 is worth 6 thousandths (0.006)

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Ordering Numbers

To order numbers (including decimals and negative numbers), compare digit by digit from the largest place value.

Worked Example: Ordering Decimals

Put these numbers in ascending order: 0.45, 0.405, 0.5, 0.045

Step 1: Write each number with the same number of decimal places by adding trailing zeros:

Number	Rewritten
0.45	0.450
0.405	0.405
0.5	0.500
0.045	0.045

Step 2: Compare as whole numbers: 045, 405, 450, 500

Answer: 0.045, 0.405, 0.45, 0.5

Ordering Negative Numbers

Remember that negative numbers further from zero are smaller. On a number line, numbers increase from left to right.

Order these from smallest to largest: -3, 5, -7, 2, -1

Answer: -7, -3, -1, 2, 5

Exam Tip: Draw a quick number line if you are unsure about ordering negative numbers. It takes a few seconds and prevents silly mistakes.

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Rounding to Decimal Places (d.p.)

To round to a given number of decimal places:

1. Count that many digits after the decimal point.
2. Look at the next digit (the "deciding digit").
3. If it is 5 or more, round up. If it is less than 5, round down.

Worked Example

Round 3.4572 to 2 decimal places.

• The second decimal place digit is 5.
• The deciding digit (third decimal place) is 7, which is 5 or more.
• Round up: 3.46

Common Rounding to Decimal Places

Original	1 d.p.	2 d.p.	3 d.p.
4.6738	4.7	4.67	4.674
12.3451	12.3	12.35	12.345
0.9961	1.0	1.00	0.996

Exam Tip: If a question says "give your answer to 2 d.p.", always write two digits after the decimal point, even if the last is zero. Writing 3.10 is correct for 2 d.p.; writing 3.1 would lose the mark.

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Rounding to Significant Figures (s.f.)

Significant figures count from the first non-zero digit.

Rules for Identifying Significant Figures

1. The first significant figure is the first non-zero digit reading from left to right.
2. All subsequent digits (including zeros) are significant until you stop counting.
3. Leading zeros are never significant.

Worked Example

Round 0.004629 to 2 significant figures.

• The first significant figure is 4 (ignore the leading zeros).
• The second significant figure is 6.
• The deciding digit is 2, which is less than 5 — round down.
• Answer: 0.0046

More Examples

Original	1 s.f.	2 s.f.	3 s.f.
3,482	3,000	3,500	3,480
0.07253	0.07	0.073	0.0725
45,678	50,000	46,000	45,700

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Truncation

Truncation means cutting off digits without rounding. You simply remove the unwanted digits.

Worked Example

Truncate 7.836 to 2 decimal places.

• Keep the first two decimal digits: 7.83
• Do not round up, even though the next digit is 6.
• Answer: 7.83

The key difference:

• Rounding 7.836 to 2 d.p. gives 7.84 (round up because 6 >= 5)
• Truncating 7.836 to 2 d.p. gives 7.83 (just cut off)

Exam Tip: Truncation is a less common topic but has appeared in recent AQA exams. Read the question carefully — if it says "truncate", do NOT round.

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Estimation

Estimation means rounding each number to one significant figure and then performing the calculation. This gives an approximate answer.

Worked Example

Estimate the value of (4.87 x 21.3) / 0.246

Step 1: Round each number to 1 significant figure:

• 4.87 rounds to 5
• 21.3 rounds to 20
• 0.246 rounds to 0.2

Step 2: Calculate:

• (5 x 20) / 0.2
• = 100 / 0.2
• = 500

When to Use Estimation

• Checking whether your calculator answer is reasonable.
• Non-calculator paper questions that say "estimate".
• Questions involving complex expressions where exact calculation is not required.

graph TD
    A[Estimation Steps] --> B[Round each value to 1 s.f.]
    B --> C[Perform the calculation]
    C --> D[State your approximate answer]
    D --> E[Check: is your answer reasonable?]

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Using Inequality Symbols

You need to know and use these symbols confidently:

Symbol	Meaning
<	Less than
>	Greater than
<=	Less than or equal to
>=	Greater than or equal to

Example: Write the correct symbol between 0.67 and 2/3.

Since 2/3 = 0.6666... (recurring), and 0.67 > 0.6666..., the answer is: 0.67 > 2/3

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Summary

• Place value determines the value of each digit based on its position.
• To order decimals, write them with the same number of decimal places and compare.
• Rounding to d.p. — count decimal places, then check the deciding digit.
• Rounding to s.f. — count from the first non-zero digit.
• Truncation cuts off digits without rounding.
• Estimation uses 1 s.f. rounding to give approximate answers.
• Always show your rounded values when estimating — marks are given for the method.

Exam Tip: In estimation questions, always show your rounded values before calculating. The marks are for the rounding process, not just the final answer.